# „Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).“

Fonte: The Number-System of Algebra, (1890), p. 86; Reported in Moritz (1914, 282)

Última atualização 4 de Junho de 2020. História
1858 - 1928

### „The concept of 'number' in its most elementary sense as the signless integer appears to be an immediate abstraction from quantitative reality subjected to processes of counting and measurement. Vulgar fractions arise from division of a quantity into equal parts. But in what sense is zero a number? Are there negative numbers? Are there numbers corresponding to incommensurable ratios? Each question requires for its solution a fresh exercise of that kind of creative imagination which we call mathematical abstraction.“

—  George Frederick James Temple British mathematician 1901 - 1992

100 Years of Mathematics: a Personal Viewpoint (1981)

### „The crowd is a social phenomenon only in the sense that it affects a number of persons at the same time.“

—  Everett Dean Martin 1880 - 1941

Fonte: The Behavior of Crowds (1920), p. 7

### „It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. …Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.“

—  Thomas Little Heath British civil servant and academic 1861 - 1940

Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)

### „When we went to school we were told that we were governed by laws, not men. As a result of that, many people think there is no need to pay any attention to judicial candidates because judges merely apply the law by some mathematical formula and a good judge and a bad judge all apply the same kind of law. The fact is that the most important part of a judge's work is the exercise of judgment and that the law in a court is never better than the common sense judgment of the judge that is presiding.“

—  Robert H. Jackson American judge 1892 - 1954

Reported in Eugene Gerhart, America's Advocate: Robert H. Jackson (1958), p. 289

### „The strengthening of behavior which results from reinforcement is appropriately called "conditioning". In operant conditioning we "strengthen" an operant in the sense of making a response more probable or, in actual fact, more frequent.“

—  B.F. Skinner American behaviorist 1904 - 1990

Science and Human Behavior (1953).

### „Common sense is in spite of, not the result of, education.“

—  Victor Hugo French poet, novelist, and dramatist 1802 - 1885

### „Simple methods will soon lead us to results of far reaching theoretical and practical importance. We shall encounter theoretical conclusions which not only are unexpected but actually come as a shock to intuition and common sense. They will reveal that commonly accepted notions concerning chance fluctuations are without foundation and that the implications of the law of large numbers are widely misconstrued.“

—  William Feller Croatian-American mathematician 1906 - 1970

Fonte: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter III, Fluctuations In Coin Tossing And Random Walks, p. 67.

### „Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.“

—  Bertrand Russell logician, one of the first analytic philosophers and political activist 1872 - 1970

Principles of Mathematics (1903), p. 451
1900s

### „Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations.“

—  Tobias Dantzig American mathematician 1884 - 1956

p, 125
Number: The Language of Science (1930)

### „The transfinite numbers are in a certain sense themselves new irrationalities“

—  Georg Cantor mathematician, inventor of set theory 1845 - 1918

As quoted in Understanding the Infinite (1994) by Shaughan Lavine
Contexto: The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.

### „With regard to the principle [of using symbols to represent sounds], it matters little whether the symbol is an elaborately detailed picture, a slightly stylized drawing, or a drastically abbreviated symbol of essentially abstract form. What is crucial is to recognize that the diverse forms perform the same function in representing sound. To see that writing has the form of pictures and to conclude that it is pictographic is correct in only one sense -- that of the form, but not the function, of the symbols. We can put it this way:QUESTION: When is a pictograph not a pictograph?ANSWER: When it represents a sound.“

—  John DeFrancis, livro The Chinese Language: Fact and Fantasy

The Chinese Language: Fact and Fantasy (1984, p. 140) http://www.pinyin.info/readings/texts/ideographic_myth.html
The Chinese Language: Fact and Fantasy (1984)

### „Since it is only in the context of a proposition that words have any meaning, our problem becomes this: To define the sense of a proposition in which a number-word occurs.“

—  Gottlob Frege mathematician, logician, philosopher 1848 - 1925

Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas. Es wird also darauf ankommen, den Sinn eines Satzes zu erklären, in dem ein Zahlwort vorkommt.
Gottlob Frege (1950 [1884]). p. 73

### „I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission.“

—  Carl Eckart American physicist 1902 - 1973

Fonte: Our Modern Idol: Mathematical Science (1984), p. 95.

### „The phenomena which we call the crowd-mind, instead of being the outgrowth of the directly social, are social only in the sense that all mental life has a social significance; they are rather the result of forces hidden in the personal and unconscious psyche of the members of the crowd, forces which are merely released by social gatherings of a certain sort.“

—  Everett Dean Martin 1880 - 1941

Fonte: The Behavior of Crowds (1920), p. 7

### „An election cannot give a country a firm sense of direction if it has two or more national parties which merely have different names but are as alike in their principles and aims as peas in the same pod.“

—  Franklin D. Roosevelt 32nd President of the United States 1882 - 1945

1930s, Fireside Chat in the night before signing the Fair Labor Standards Act (1938)

### „Marcel Duchamp's 'Nude descending the stairs' is the result of the desire for motion. Here he has also eliminated representative form. This avoids the connotation of ideas which would interfere with the success of the main issue - the sense of movement.“

—  Alexander Calder American artist 1898 - 1976

1930s - 1950s, Statement from Modern Painting and Sculpture', (1933)
Fonte: en.wikiquote.org - Alexander Calder / Quotes of Alexander Calder / 1930s - 1950s / Statement from Modern Painting and Sculpture', (1933)

### „Perhaps the least inadequate description of the general scope of modern Pure Mathematics—I will not call it a definition—would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.“

—  E. W. Hobson British mathematician 1856 - 1933

Fonte: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 287; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/4/mode/2up, (1914), p. 5: Definitions and objects of mathematics.

### „Speculative philosophy as the realisation of God is the positing of God, and at the same time his cancellation or negation; theism and at the same time atheism: for God – in the sense of theology – is God only as long as he is taken to be a being distinguished from and independent of the being of man as well as of nature. The theism that as the positing of God is simultaneously his negation or, conversely, as the negation of God equally his affirmation, is pantheism. Theological theism – that is, theism properly speaking – is nothing other than imaginary pantheism which itself is nothing other than real and true theism.“

—  Ludwig Feuerbach German philosopher and anthropologist 1804 - 1872

Part I, Section 14
Principles of Philosophy of the Future http://www.marxists.org/reference/archive/feuerbach/works/future/index.htm (1843)

### „The hospitality of primitive peoples, respect for human life, the sense of reciprocal obligation, compassion for the weak, courage, extending even to the sacrifice of self for others which is first learnt for the sake of children and friends, and later for that of members of the same community — all these qualities are developed in man anterior to all law, independently of all religion, as in the case of the social animals.“

—  Peter Kropotkin Russian zoologist, evolutionary theorist, philosopher, scientist, revolutionary, economist, activist, geographer, writer 1842 - 1921

Fonte: Law and Authority (1886), II
Contexto: The hospitality of primitive peoples, respect for human life, the sense of reciprocal obligation, compassion for the weak, courage, extending even to the sacrifice of self for others which is first learnt for the sake of children and friends, and later for that of members of the same community — all these qualities are developed in man anterior to all law, independently of all religion, as in the case of the social animals. Such feelings and practices are the inevitable results of social life. Without being, as say priests and metaphysicans, inherent in man, such qualities are the consequence of life in common.
But side by side with these customs, necessary to the life of societies and the preservation of the race, other desires, other passions, and therefore other habits and customs, are evolved in human association. The desire to dominate others and impose one's own will upon them; the desire to seize upon the products of the labor of a neighboring tribe; the desire to surround oneself with comforts without producing anything, while slaves provide their master with the means of procuring every sort of pleasure and luxury — these selfish, personal desires give rise to another current of habits and customs.

### „Everything rhythmically organic is true. Everything, which results from the proper feeling for rhythmically organized spiritual units, is true and alive — alive within itself. When we lose the sense for such true beauty we lose our natural sense for the rich flavor of life, which is the basis for all inspirational work.Things generally taken for beautiful are nothing other than the product of frozen, stereotyped taste, bound by sterile rules and purely exterior judgment.“

—  Hans Hofmann American artist 1880 - 1966

'Painting and Culture' p. 57
Search for the Real and Other Essays (1948)